Maxwell's equations in the vacuum with electric permittivity $\epsilon_0$ and magnetic permeability $\mu_0$ are given as:

$$\nabla \cdot \vec E = \frac{\rho}{ \epsilon_0}$$ $$\nabla \cdot \vec B = 0$$ $$\nabla \times \vec E = - \frac {\partial \vec B}{\partial t}$$ $$\nabla \times \vec B = \mu_0 \vec J + \epsilon_0 \mu_0 \frac {\partial \vec E}{\partial t}$$

In material media, $\epsilon$ and $\mu$ are larger or smaller than $\epsilon_0$ and $\mu_0$ and may depend on $\vec x$ and even on the direction of polarization.

All that seems okay to me at first glance. However, in nonlinear media, $\epsilon$ and $\mu$ depend on $\vec E$ and $\vec B$. So, for nonlinear media, Maxwell's equations are often written as:

$$\nabla \cdot \vec E = \frac{\rho}{ \epsilon (\vec E,\vec B)}$$ $$\nabla \cdot \vec B = 0$$ $$\nabla \times \vec E = - \frac {\partial \vec B}{\partial t}$$ $$\nabla \times \vec B = \mu (\vec E,\vec B) \vec J + \epsilon (\vec E,\vec B) \mu (\vec E,\vec B) \frac {\partial \vec E}{\partial t}$$

(As a further generalization, $\epsilon$ and $\mu$ are sometimes represented as tensors whose components are functions of $\vec E$ and $\vec B$, but that is not an important issue for the current question.)

My problem with the nonlinear material media version of Maxwell's equations is that it seems to assume instantaneous material response to changing $\vec E$ and $\vec B$, while it seems that any physically plausible material can only respond in finite time. It would be like saying that a spring's length is proportional to the force applied- which is true only when the applied force is changed very slowly. That is, I expect that any real material will have a dynamic response to changing $\vec E$ and $\vec B$.

If that's true, then it seems it should make better sense for the specification of $\epsilon$ and $\mu$ to be in the form of differential or integral equations including time. Of course that would complicate the math a lot, but from a physics perspective it would be more plausible. My question: Is there a form of Maxwell's equations in nonlinear media that takes the dynamic response of the medium into account? A follow-up question would be "Is there a Lorentz-covariant form of those equations?"

  • $\begingroup$ Maxwell's equation account for the dynamical response by allowing the refractive index to be a complex number. This allows for a phase difference between the external field and the polarisation response. This holds for linear and nonlinear response. $\endgroup$
    – my2cts
    Commented Mar 6, 2019 at 18:25
  • $\begingroup$ My understanding is that a complex refractive index allows for absorption and amplification of EM waves passing through the medium. I don't see how it would be enough to account for a more general sort of dynamic response. $\endgroup$
    – S. McGrew
    Commented Mar 6, 2019 at 21:20
  • $\begingroup$ That is hard to argue with. Can you give an example where it would fail? $\endgroup$
    – my2cts
    Commented Mar 6, 2019 at 21:38
  • $\begingroup$ Are you asking if I can give a counterexample to the idea that a complex refractive index allows for absorption and amplification of EM waves passing through the medium? $\endgroup$
    – S. McGrew
    Commented Mar 6, 2019 at 21:53
  • $\begingroup$ I think a complex refractive index does not make Maxwell's equations nonlinear, so does not describe electromagnetism in a nonlinear medium. $\endgroup$
    – S. McGrew
    Commented Mar 6, 2019 at 22:07

2 Answers 2


What you are talking about is dispersion. Dispersion is not necessarily nonlinear phenomenon, it occurs in linear media as well. Moreover you can have spatial and temporal dispersion. Temporal dispersion means that system response depends on what is the stimulus is currently as well as on what it was earlier. Spatial dispersion means that you material response at position A depends on what the field does at position $B\neq A$

There are many ways to account for these phenomena, I will only list how it is done in dielectrics. Other generalizations are similar

In non-trivial dielectics you would have





Now you can stick all you complex material response into displacement. Want temporal dispersion (linear case)? Here you go:

$\mathbf{D}\left(t,\mathbf{r}\right)=\epsilon_0\int^t_{-\infty}dt' \boldsymbol{\epsilon_r}\left(t-t',\mathbf{r}\right).\mathbf{E}\left(t',\mathbf{r}\right)$

Spatial dispersion (linear)?

$\mathbf{D}\left(t,\mathbf{r}\right)=\epsilon_0\int d^3r' \,\boldsymbol{\epsilon_r}\left(t,\mathbf{r}-\mathbf{r'}\right).\mathbf{E}\left(t,\mathbf{r'}\right)$

For non-linear response you play similar games but you tend to use polarization density, i.e. $\mathbf{P}=\mathbf{D}-\epsilon_0\mathbf{E}$. Second order non-linearity with temporal dispersion:

$\mathbf{P}\left(t,\mathbf{r}\right)=\int^t_{-\infty} dt'\int^t_{-\infty} dt'' \boldsymbol{\chi^{(2)}\left(t-t',t-t'',\mathbf{r}\right)}.\mathbf{E}\left(t',\mathbf{r}\right).\mathbf{E}\left(t'',\mathbf{r}\right)$

etc... Most books on nonlinear optics will cover this

PS: $\epsilon_r$ is the relative perimittivity tensor, $\chi^{(2)}$ is the second-order susceptibility tensor.

  • $\begingroup$ Actually, I am specifically not asking about dispersion. My interest is in how to describe field/material nonlinearities in a Lagrangian-derived, covariant fashion. $\endgroup$
    – S. McGrew
    Commented Mar 6, 2019 at 23:15
  • $\begingroup$ Sorry, my bad. Well, can you not go with a similar route, but instead of polarization you would have a polarization-magnetization asymmetric rank-2 tensor, which is a function of the electromagnetic tensor? Once you have done this one can investigate how to bring the disperion into it, but you will probably end up with some transfer function integrating over the 'past' half of the light-cone for the point of spacetime in question. I am sorry for being vague, but this is not something I have yet encountered (apart from the polarization-magnetization tensor). $\endgroup$
    – Cryo
    Commented Mar 6, 2019 at 23:30
  • $\begingroup$ I've done a lot of literature searching, but it seems that maybe this subject is complicated enough to discourage most theoreticians. Most of the relevant papers have a very limited focus, such as only nonlinear permittivity, or moving media, or rubber magnets with magnetic nanoparticles embedded in them ; and the ones that at first seem most relevant often ignore details like relativistic invariance. $\endgroup$
    – S. McGrew
    Commented Mar 7, 2019 at 0:04
  • $\begingroup$ I would be interested to see a literature on covariant nonlinear optics if you find it. Are you developing the theoretical treatment yourself, or do you need a verified publication? If it is ther former, I think you should start working with polarization-magnetization tensor, which can plug in easily into the Lagrangian density. Unfortunatelly I have no expretise here to give advice I can trust to be good. $\endgroup$
    – Cryo
    Commented Mar 7, 2019 at 0:20
  • $\begingroup$ I think I'm poking around in under-explored territory. Will be glad to share whatever I can find. $\endgroup$
    – S. McGrew
    Commented Mar 7, 2019 at 1:04

Is there a form of Maxwell's equations in nonlinear media that takes the dynamic response of the medium into account?

Yes there is, but it probably won't satisfy you. The general form is the same as the usual Maxwell's equations for fields in the presence of known charge and current distribution in vacuum. The only thing the material medium is assumed to change is that distributions $\rho,\mathbf j$ have another contribution due to material medium.

Such Maxwell's equations are not a complete system of differential equations, it is instead an underspecified system, so some other assumptions must be introduced and used to relate distributions of charge and current on one side, and EM fields on the other side.

These assumptions vary with the physical situation such as static dielectric polarization ($\mathbf P$ is sufficient), static ferromagnet magnetization ($\mathbf M$ and $\mathbf j_{free}$ is sufficient), or high frequency dissipative EM wave propagation (one better works with some microscopic model and $\rho,\mathbf j$ directly). They also depend on the quality of the material medium, which there are a lot of kinds. There is no general formulation of EM theory of material medium that would provide a closed system of equations.

  • $\begingroup$ That's what I've been suspecting. Are there specific conceptual or mathematical difficulties that have prevented such a formulation? If so, you know of any papers discussing the difficulties? $\endgroup$
    – S. McGrew
    Commented Mar 7, 2019 at 3:43
  • $\begingroup$ @S.McGrew charge and current distribution in macroscopic matter depend on what very high number of constituent particles do. To obtain a formulation of the kind you seek, there would have to be universal simplification of high number of microscopic variables into a few macroscopic variables. Nobody has found a universal way to do that, there are only situation-adapted formulations of limited validity, as described in my answer. There are papers and books on derivations of macroscopic EM theory from microscopic models, classical and quantum-theoretical. $\endgroup$ Commented Mar 7, 2019 at 12:47
  • $\begingroup$ This was studied much by Netherlandian physicists, see, for example, S. R. de Groot, J. Vlieger, Derivation of Maxwell's equations: The statistical theory of the macroscopic equations, Physica 31, 254-268 doi.org/10.1119/1.1976000 $\endgroup$ Commented Mar 7, 2019 at 12:54
  • 1
    $\begingroup$ @JánLalinský Thanks for the reference. The general coarse graining procedure is part and parcel of any good text book (e. g. in Chapter 6.6 of Jackson's Classical Electrodynamics). $\endgroup$
    – Max Lein
    Commented Mar 8, 2019 at 5:39

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